Let's find the coordinates of point [tex]\( P \)[/tex] which divides the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] in the ratio of [tex]\( 2:1 \)[/tex].
Given:
- Point [tex]\( A \)[/tex] has coordinates [tex]\((2, -1)\)[/tex]
- Point [tex]\( B \)[/tex] has coordinates [tex]\((4, -3)\)[/tex]
- The ratio [tex]\( m:n = 2:1 \)[/tex]
We need to use the section formula to find the coordinates of [tex]\( P \)[/tex]:
[tex]\[
P(x, y) = \left( \frac{m \cdot x_2 + n \cdot x_1}{m + n}, \frac{m \cdot y_2 + n \cdot y_1}{m + n} \right)
\][/tex]
Here [tex]\( m = 2 \)[/tex], [tex]\( n = 1 \)[/tex], [tex]\( x_1 = 2 \)[/tex], [tex]\( y_1 = -1 \)[/tex], [tex]\( x_2 = 4 \)[/tex], and [tex]\( y_2 = -3 \)[/tex].
First, let's find the [tex]\( x \)[/tex]-coordinate of [tex]\( P \)[/tex]:
[tex]\[
x = \frac{m \cdot x_2 + n \cdot x_1}{m + n}
\][/tex]
[tex]\[
x = \frac{2 \cdot 4 + 1 \cdot 2}{2 + 1}
\][/tex]
[tex]\[
x = \frac{8 + 2}{3}
\][/tex]
[tex]\[
x = \frac{10}{3}
\][/tex]
[tex]\[
x = 3.333333333333333
\][/tex]
Next, let's find the [tex]\( y \)[/tex]-coordinate of [tex]\( P \)[/tex]:
[tex]\[
y = \frac{m \cdot y_2 + n \cdot y_1}{m + n}
\][/tex]
[tex]\[
y = \frac{2 \cdot (-3) + 1 \cdot (-1)}{2 + 1}
\][/tex]
[tex]\[
y = \frac{-6 - 1}{3}
\][/tex]
[tex]\[
y = \frac{-7}{3}
\][/tex]
[tex]\[
y = -2.333333333333333
\][/tex]
Thus, the coordinates of point [tex]\( P \)[/tex] are [tex]\( (3.333333333333333, -2.333333333333333) \)[/tex].
